Stopped Compound Poisson Process and Related Distributions
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This chapter considers the first-crossing problem of a compound Poisson process with positive integer-valued jumps in a nondecreasing lower boundary. The cases where the boundary is a given linear function, a standard renewal process, or an arbitrary deterministic function are successively examined. Our interest is focused on the exact distribution of the first-crossing level (or time) of the compound Poisson process. It is shown that, in all cases, this law has a simple remarkable form which relies on an underlying polynomial structure. The impact of a raise of a lower deterministic boundary is also discussed.
Keywords and phrasesCompound Poisson process first-crossing lower boundary ballot theorem generalized Abel-Gontcharoff polynomials generalized Poisson distribution quasi-binomial distribution damage model
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